Golden ratio

I believe the geometric proportion served the Creator as an idea
when He introduced the continuous generation of similar
objects from similar objects. -- Johannes Kepler

The Golden Ratio can be obtained easily by the proper division of a line segment,
instead of constructing the geometrically pleasing rectangles. Simply
divide the line segment into two parts of uneven lengths, so that the
ratio of the smaller part to the larger part equals the ratio of
the larger part to the entire line segment.

Following on from what mblase said: You can draw a series of squares inside a 'golden ratio rectangle' that theoretically goes on forever. If you keep turning the paper clockwise, and then always draw the square on the left, you can then draw a spiral through the diagonally opposite corners of the squares. This is known as the golden spiral.

This particular spiral is yet another example of the golden ratio in nature; many of the most attractive seashells describe this shape.

Although, let me offer a disclaimer here; just because a fair amount of natural shapes fit the golden ratio, this does not give it some kind of magical or religious significance. It could well be that a lot of seashells adopt the 'golden ratio spiral' simply because it is the easiest to build - after all, DNA is known to often create things with a fractal or recursive nature, and the golden ratio is recursive by definition.

Still, a lot of things need to be learnt about the golden ratio. Why do we find it so attractive? Is it because our faces, too, fit this ratio? (With our eyebrows and chin forming the square within the recangle, might I add). Or is this just coincedence? Don't expect an answer anytime soon...

...which is approximately 1.618(03398874989484820458683436563811...). Φ-1 (the inverse of Φ) is common enough to receive its own symbol: φ (lower-case phi).

The ancient Egyptians believed that this "sacred ratio" was important enough to embed in their art and constructions. Many Egyptian temples employ rectangluar archways designed according to the Golden Ratio. At the Great Pyramid of Giza, the ratio of the length of one side of the base to the perpendicular height of the pyramid is about 2/√Φ, making the slant height of the pyramid side proportionately equal to Φ. The result is that each side of the pyramid is a Golden Triangle.

A Golden Triangle is similar to a Golden Rectangle in its behavior. It's an isoceles triangle with angles measuring 36°, 72° and 72°. It can be created from a regularpentagon by drawing lines from any vertex to the two vertices opposite it. If the base of this triangle (the short side) is Φ units long, the other two sides are 1+Φ units long. By bisecting one base angle of the triangle, two more isoceles triangles are produced and the smaller one is another Golden Triangle:

More famously, Aristotle and the ancient Greeks believed that rectangles possessing the Golden Ratio were inherently aesthetic. The Parthenon, for example, is constructed in such a way that the front of the temple is exactly contained in a Golden Rectangle, and the "dividing line" mentioned above lying on either side of the entryway.

Drawing these Golden figures by hand is difficult, since the Golden Ratio is an irrational number, a never-ending decimal. However, it's fairly easy to draw it using a straightedge and compass. First draw a short vertical line segment (of, say, length 1) and at one end draw a second segment perpendicular to it which is twice as long (length 2):

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Connect the endpoints to form a right triangle; the hypotenuse of this triangle is of length √5. Draw an arc centered at the point where the short side meets the hypotenuse, dividing the hypotenuse into two segments of length 1 and √5-1. Finally, draw a second arc centered at the point where the longer side meets the hypotenuse, with the radius equal to the longer section of the hypotenuse, and mark where it crosses the longer side:

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The longer side is divided into two lengths, one measuring √5-1 and the other 2-(√5-1) = 3-√5. The ratio of the first number to the second is exactly (1+√5)/2 -- the Golden Ratio, Φ.

Armed with this, we can now illustrate the Golden Ratio using a single line segment divided in two, known as a Golden Section. Rewriting the equation Φ2-Φ-1=0 tells us Φ2 = Φ+1, or visually:

(It's an interesting property of the Golden Ratio that Φ-1 = 1/Φ and Φ+1 = Φ2.) The ratio of 1 to Φ is algebraically equal to the ratio of Φ to Φ2, and so the geometric definition of the Golden Ratio is preserved: "the smaller is to the larger, what the larger is to the whole."

However, it's more interesting to see the implications of the Golden Ratio when rectangles are used. Since the Ratio remains constant no matter what the actual size of the rectangle is, the smaller rectangle is just as Golden as the larger one. We can subtract another square from it, and so continue ad infinitum:

If you begin from the lower-left vertex of the largest rectangle and draw a spiral through the vertex of every smaller square, you can continue until the squares become infinitely small. Interestingly, you can create the same spiral by tracing the vertices of a series of Golden Triangles.

This is not entirely a coincidence, because the Golden Ratio is also closely tied to the Fibonacci numbers. A sequence of Fibonacci numbers is constructed by beginning with any two numbers (but typically 1 and 1) and adding them to produce a third, then adding the second and third to produce the fourth, and so on: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. As this sequence continues, the ratio between any number and the number before it rapidly approaches the Golden Ratio: 34/21 = 1.6190..., 55/34 = 1.6176..., and 89/55 = 1.6182....

We may represent the Fibonacci sequence geometrically by drawing two squares with sides 1 unit in length side by side, forming a rectangle of sides 1 and 2. Add a 2-unit square beside this to form a rectangle of sides 2 and 3. Add a 3-unit square to make a 3-by-5 rectangle, a 5-unit square to make a 5-by-8 rectangle, and so on in a spiral shape. As the squares get larger and larger and the sequence continues, the rectangles become more and more Golden and the spiral more and more logarithmic:

The logarithmic spiral is not limited to non-human elements in nature. As a human embryo grows, its spine unfolds in a pattern very similar to the logarithmic spiral. This is not the only place the Golden Ratio appears in the human body, however. Leonardo da Vinci, Renaissance man that he was, noticed this and used it in his art. The well-known Vitruvian Man employs the Golden Ratio extensively; for instance, the distances between the top of his head, the bottom of his feet, and his navel between them lie on a perfect Golden Section. The face of the Mona Lisa can be neatly inscribed in a Golden Rectangle, and the positions of her eyes, nose and mouth are also placed according to the Golden Ratio.

It is sometimes said that the "most aesthetically pleasing" human face and body relies entirely on the Golden Ratio, and that this is the reason it is so aesthetically satisfying in art and architecture as well.

This information is common mathematical knowledge, but my primary sources included:
The "Phi-Nest" (http://goldennumber.net)
The Golden Mean (http://galaxy.cau.edu/tsmith/KW/golden.html)